Fermi Liquid Behavior of the In-Plane Resistivity in the Pseudogap

Home , Cuprate superconductor

Fermi liquid behavior of the in-plane resistivity in the

pseudogap state of YBa 2Cu 4O8

Cyril Proust 1*, B. Vignolle 1, J. Levallois 1,2, S. Adachi 3 and N. E. Hussey 4,5*

1 Laboratoire National des Champs Magnétiques Intenses (LNCMI-EMFL), (CNRS-INSA- UGA-UPS), Toulouse 31400, France.

2 Department of Quantum Matter Physics, University of Geneva, CH-1211 Geneva 4, Switzerland.

3 Superconductivity Research Laboratory, Shinonome 1-10-13, Koto-ku, Tokyo 135-0062, Japan.

4 High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525ED Nijmegen, Netherlands

5 Radboud University, Institute of Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, Netherlands

* To whom correspondence may be addressed. Email: [email protected] or [email protected]

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Our knowledge of the ground state of underdoped hole-doped cuprates has evolved considerably over the last few years. There is now compelling evidence that inside the pseudogap phase, charge order breaks translational symmetry leading to a reconstructed Fermi surface made of small pockets. Quantum oscillations, (Doiron-Leyraud N, et al. (2007) Nature 447:564-568), optical conductivity (Mirzaei SI, et al. (2013) Proc Natl Acad Sci USA 110:5774-5778) and the validity of Wiedemann-Franz law (Grissonnache G, et al. (2016) Phys. Rev. B 93:064513) point to a Fermi liquid regime at low temperature in the underdoped regime. However, the observation of a quadratic temperature dependence in the electrical resistivity at low temperatures, the hallmark of a Fermi liquid regime, is still missing. Here, we report magnetoresistance measurements in the magnetic- 2 field-induced normal state of underdoped YBa 2Cu 4O8 which are consistent with a T resistivity extending down to 1.5 K. The magnitude of the T2 coefficient, however, is much smaller than expected for a single pocket of the mass and size observed in quantum oscillations, implying that the reconstructed Fermi surface must consist of at least one additional pocket.

Significance

High temperature superconductivity evolves out of a metallic state that undergoes profound changes as a function of carrier concentration, changes that are often obscured by the high upper critical fields. In the more disordered cuprate families, field suppression of superconductivity has uncovered an underlying ground state that exhibits unusual localization behavior. Here, we reveal that in stoichiometric YBa 2Cu 4O8, the field-induced ground state is both metallic and Fermi-liquid like. The manuscript also demonstrates the potential for using the absolute magnitude of the electrical resistivity to constrain the Fermi surface topology of correlated metals and in the case of YBa 2Cu 4O8, reveals that the current picture of the reconstructed Fermi surface in underdoped cuprates as a single, isotropic electron-like pocket may be incomplete.

Introduction

The generic phase diagram of Fig. 1 summarizes the temperature and doping dependence of the 1 in-plane resistivity ρab (T) of hole-doped cuprates . Starting from the heavily overdoped side, non-superconducting La 1.67Sr 0.33CuO 4 for example shows a purely quadratic resistivity below ~ 50 K (ref. 2). Below a critical doping pSC where superconductivity sets in, ρab (T) exhibits 2 n supra-linear behaviour that can be modeled either as ρab ~ T + T or as T (1 < n < 2). When a magnetic field is applied to suppress superconductivity on the overdoped side, the limiting low- T behavior is found to be T-linear 3,4,5. Optimally doped cuprates are characterized by a linear resistivity for all T > Tc, though the slope often extrapolates to a negative intercept suggesting 1 that at the lowest temperatures, ρab (T) contains a component with an exponent larger than one . In the underdoped regime, ρab (T) varies approximately linearly with temperature at high T, but as the temperature is lowered below the pseudogap temperature T*, it deviates from linearity in a very gradual way 6. At lower temperatures, marked by the light blue area in Fig. 1, there is now compelling evidence from various experimental probes of incipient charge order 7-11 . High 12,13 14 field NMR and ultrasonic measurements indicate that a phase transition occurs below Tc. This is also confirmed by recent high field x-ray measurements which indicate that the CDW order becomes tridimensional with a coherence length that increases with increasing magnetic field strength 15,16 . This leads to a Fermi surface reconstruction that can be reconciled with quantum oscillations 17,18 as well as with the sign change of the Hall 19 and Seebeck 20 coefficients. Whether the charge order is biaxial 21 or uniaxial with orthogonal domains 22 is still 3 an open issue, but a Fermi surface reconstruction involving two perpendicular wavevectors leads to at least one electron pocket in the nodal region of the Brillouin zone 23 . Depending on the initial pseudogapped Fermi surface and on the wavevectors of the charge order, Fermi surface reconstruction can also lead to additional, smaller hole pockets24,25 . In Y123 (doping level p ≈ 0.11) the quantum oscillation spectra consist of a main frequency Fa = 540 T and a beat pattern indicative of nearby frequencies Fa2 = 450 T and Fa3 = 630 T. A smaller frequency Fh ≈ 100 T has been detected by thermopower and c-axis transport measurements and attributed 26 there to an additional small hole pocket . The presence of the three nearby frequencies Fai can be explained by a model involving a bilayer system with an electron pocket in each plane and 27,28 magnetic breakdown between the two pockets . In this scenario, the low frequency Fh could originate from quantum interference or the Stark effect 29 . However, this scenario predicts the occurrence of five nearby frequencies, and thus requires fine tuning of certain microscopic parameters such as the bilayer tunneling t⊥. Moreover, the doping dependence of the Seebeck coefficient is difficult to reconcile with a Fermi surface reconstruction scenario leading to only one electron pocket per plane. More generally, the observation of quantum oscillations is a classic signature of Landau quasiparticles. In underdoped Y123, the temperature dependence of the amplitude of the oscillations follows Fermi-Dirac statistics up to 18 K, as in the Landau-Fermi liquid theory 30 . This conclusion is supported by other observations, such as the validity of the Wiedemann- Franz law 31 in underdoped Y123 and the quadratic frequency and temperature dependence of the quasiparticle lifetime τ (ω, T) measured by optical spectroscopy32 in underdoped HgBa 2CuO 4+ δ (Hg1201). An important outstanding question is whether the in-plane resistivity of underdoped cuprates also exhibits the behavior of a canonical Landau-Fermi liquid, namely a quadratic temperature dependence at low T? In underdoped cuprates, several studies have 2 shown ρab ~ T , but always at elevated temperatures (indeed, most are above Tc) and only over a limited temperature range that never exceed a factor of 2.5 6,33 -36 . (see Table S1 for a detailed 36 list of studies). At low temperatures, either ρab (T) starts to become non metallic , suggesting that the T 2 behavior observed at intermediate temperatures could just be a crossover regime, or a quadratic behaviour has previously been hinted at19 , rather than shown explicitly. Here, we present high field in-plane magnetoresistance measurements in underdoped YBa 2Cu 4O8 (Y124) 2 that are consistent with the form ρa(T) = ρ0(T) + AT from T = Tc down to temperatures as low as 1.5 K, e.g. over almost two decades in temperature. In addition, we investigate the magnitude of the resultant A coefficient and compare it with some of the prevalent Fermi surface reconstruction scenarios. In conclusion, we find that the magnitude of A is difficult to reconcile with the existence of a single electron pocket per plane, with an isotropic mass.

Results

We have measured the a-axis magnetoresistance (i.e. perpendicular to the conducting CuO chains) of two underdoped Y124 (Tc = 80 K) single crystals up to 60 T at various fixed temperatures down to 1.5 K. From thermal conductivity measurements at high fields, the upper 37 critical field Hc2 of Y124 has been estimated to be ~ 45 T . Raw data for both samples are shown in Fig. 2. Above Tc, i.e. in the absence of superconductivity, the transverse magnetoresistance can be accounted for, over the entire field range measured , by a two-carrier model 35 using the formula: (1) () = (0) + where ρ(0) is the zero-field resistivity and α and β are free parameters that depend on the conductivity and the Hall coefficient of the electron and hole carriers 35 (see Fig. S1 for a comparison of the two-band and single-band, quadratic forms for the magnetoresistance.) To 4

obtain reliable values of ρ (H→0, T) = ρ(0) and corresponding error bars for each field sweep, the data were fitted to equation (1) in varying field ranges using the procedure described in detail in Figs. S2-S5 and Tables S2-S3. Precisely the same form is used to fit the high-field data at all temperatures studied below Tc. This procedure has been found to yield reliable ρ(0) values in both cuprate 5 and pnictide 38 superconductors. Extrapolation of the high-field data to the zero- field axis ρ(0), as shown by dashed lines in Fig. 2, allows one then to follow the evolution of ρa(T) down to low temperatures. The extrapolated ρ(0) values are plotted versus temperature in Fig. 3 (symbols) for both crystals, along with the zero-field temperature dependence of the resistivity (solid line). The dashed lines in Fig. 3 correspond to fits of the ρ(0, T) data to the 2 n form ρ0(T) = ρ0 + AT . A fit of the data to the form ρ0(T) = ρ0 + AT yields n = 1.9 + 0.2 is 2 shown in the Fig. S6. The inset of the Fig. 3 are corresponding plots of ρa(T) versus T to highlight the approximately quadratic form of ρa(T). From the dashed line fits we obtain ρ0 = 7.5 ± 1.0 and 10.0 ± 1.0 µΩ.cm and A = 10.0 ± 1 and 8.5 ± 0.5 nΩ cm K-2 for samples #1 and #2 respectively.

Discussion

The first key result of this study is our observation, within experimental resolution, of a quadratic temperature dependence of the in-plane resistivity in Y124 down to low temperatures, which indicates that the low-lying (near-nodal) electronic states inside the pseudogap phase of underdoped cuprates bear all the hallmarks of Landau quasiparticles. Intriguingly, the magnitude of the T2 term, A = 9.5 ± 1.5 nΩ cm K-2, is similar to that measured at high temperature, e.g. above 50 K in underdoped Y123 at p = 0.11 (A ≈ 6.6 nΩ cm K-2) (ref. 33) as well as in single-layer Hg1201 above 80 K where A varies between ≈ 10 and 15 nΩ cm K-2 for 0.055 < p < 0.11 (ref. 34). Note that in Y124, such comparison cannot be made since the temperature dependence of the resistivity is not quadratic above Tc. In Y123, it is known that * an incipient charge density wave (CDW) is formed below T , which onsets at about TCDW ≈ 130 - 150 K in the doping range at p = 0.11 – 0.14 ( ref. 39,40). At a lower temperature TFSR ≈ 50 K, high-field NMR 12 and ultrasound 14 measurements for p = 0.11 have revealed a phase transition, below which long range static charge order appears and Fermi surface reconstruction is believed to take place. Taking into account that the ρ(0) values shown in Fig. 3 are extrapolated from this high-field phase, there would appear to be no significant change in the A coefficient between the low-T regime where long range CDW sets in and the high-T regime where only incipient CDW order is detected. This is reminiscent of the situation in NbSe 2 where there is negligible change of the resistivity at the CDW transition TCDW = 33 K (ref. 41). This behavior can be understood if no substantial change of the Fermi surface occurs at the CDW transition (for a discussion see ref. 42). To make an analogy with Y124, we first acknowledge that the effect of the pseudogap is to suppress quasiparticles near the Brillouin zone boundaries. A Fermi surface reconstruction due to charge order with dominant wavevectors (Qx, 0) and (0, Qy) will create a small electron-like pocket composed of the residual ‘nodal’ density of states, in contrast to the large hole-like Fermi surface characterizing the overdoped state. This Fermi surface reconstruction can be seen as a folding of the Fermi surface and in terms of transport properties, the same nodal states will be involved in scattering processes below T* and below TFSR when the Fermi surface is reconstructed. Thus, the similar value of A at high and low temperatures can be reconciled. Note that in canonical 1D CDW systems such as NbSe 3 (ref. 43) and organics metals 44 , there is a marked change of slope of the resistivity below the charge ordering temperature due to nesting of part of the original Fermi surface.

The scenario discussed above for Y124 is also consistent with the observation of an anisotropic scattering rate Γ in cuprates. In overdoped Tl 2Ba 2CuO 6+ δ (Tl2201) (ref. 45), for example, it has been shown that the scattering rate is composed of two distinct terms, a T2 term that is almost isotropic within the basal plane and a T-linear scattering rate that is strongly anisotropic, vanishing along the zone diagonals and exhibiting a maximum near the Brillouin zone boundary, where the pseudogap is maximal. Inside the pseudogap regime, therefore, one expects the T-linear scattering rate to become much diminished, leaving the T2 scattering term as the dominant contribution to ρa(T). In correlated metals, both the T2 resistivity and the T-linear specific heat are the consequence of the Pauli exclusion principle. Thus, the strength of the T2 term in ρ(T) is empirically related to the square of the electronic specific heat coefficient γ0 via the Kadowaki-Woods ratio (KWR) 2 A/γ0 (ref. 46). An explicit expression for the KWR has been derived for correlated metals taking into account unit cell volume, dimensionality and carrier density 47 . In a single band quasi-2D metal, the A coefficient reads: ∗ (2) = ℏ . * where a and c are the lattice parameters, and m and kF are respectively the (isotropic) effective mass and Fermi wavevector. A similar expression has also been obtained recently using the 48 Kubo formalism . As shown in the Supplementary, comparison of AKWR with experimentally determined values for both single- and multi-band correlated oxides shows good agreement for AKWR values spanning over three orders of magnitude. Electron-electron collisions involve two quasiparticles that reside within a width of order kBT near the Fermi energy, providing the factor T2. However, the total electron momentum is conserved in normal electron-electron scattering for the simple metals with a nearly-free- electron-like Fermi surface. Additional mechanisms, such as Umklapp or interband scattering, are thus needed in order to understand the dissipation (see ref. 49-51 for a discussion). With regards the hole-doped cuprates, it is debatable whether the KWR should hold at all within the pseudogap phase. However, given the increasing amount of data pointing to a rather conventional state at sufficiently low temperature, we believe that such analysis and comparison is appropriate and in the following, we consider briefly a number of these possible mechanisms 47-51 in turn.

Umklapp scattering Assuming a single electron pocket per CuO 2 plane, Umklapp collisions can lead to dissipation for electron-electron scattering since the condition on the reconstructed FS, kF > G/4, is fulfilled in the reconstructed Brillouin zone (G is a reciprocal lattice vector). In Y124, the QO frequency -1 linked to the electron pocket (Fe = 660 ± 30 T) converts into kFe = 1.42 ± 0.03 nm (ref. 52,53), while the most recent QO measurements have indicated that m* = 1.9 ± 0.1 me (ref. 54-55). Equation (2) thus gives an estimate of the A coefficient for the electron pocket, Ae = 86 ± 20 -2 -2 nΩ cm K , i.e. AKWR = 43 ± 10 nΩ cm K taking into account the two CuO 2 planes. Significantly, this is almost five times larger than the experimental value. Assuming that the KWR ratio holds in underdoped cuprates, it implies that a reconstructed Fermi surface containing only one electron pocket (with an isotropic m*) cannot account fully for the magnitude of the T2 resistivity term in Y124. Similar conclusions are also drawn from comparison of the Fermi parameters reported for underdoped Y123 (ref. 18) and the measured A coefficient 33 (albeit at elevated temperatures).

Multiband scenario In a second scenario initially proposed by Baber, the momentum transfer between two distinct reservoirs can also lead to dissipative scattering 49 . In underdoped Y123, a small QO frequency (Fh ≈ 95 T) was discovered and attributed to a hole pocket based on the doping dependence of the Seebeck coefficient 26 . A Fermi surface comprising at most one electron and two hole pockets with the measured areas and masses is consistent with a scenario based on the Fermi- surface reconstruction induced by the CDW order observed in Y123 24 (see Fig. 4). It is also compatible, within error bars, with the value of the electronic specific heat measured at high fields in YBCO 56,57 (see Supplementary for details). (Before proceeding, it is worth noting that the multiband scenario relies on the assumption that the small oscillations detected in ref. 26 derive from a small hole pocket. Other studies have suggested that the small oscillation Fh owes its origin in quantum interference or the Stark effect 29 in a magnetic breakdown scenario between the bilayer of Y123 (ref. [23, 27, 28])). Taking the parameters for the two types of pockets in underdoped Y123 (p = 0.11): kFe= 1.28 ± -1 * -1 * 0.02 nm and m e= 1.4 ± 0.1 me, kFh = 0.54 ± 0.03 nm and m h = 0.45 ± 0.1 me, we obtain Ae -2 -2 = 56 ± 9 nΩ cm K and Ah = 89 ± 42 nΩ cm K (note that the large error here is due to the relative error in the effective mass). Finally, applying the parallel-resistor formula and taking into account the bilayer nature of Y123, the A coefficient is estimated to be AKWR ≈ 12 ± 6 nΩ cm K-2, in reasonable agreement with the value measured at high temperature (A ≈ 6.6 nΩ cm K-2) (ref. 33). CDW order has not yet been directly observed by X-ray or by NMR in Y124, but the similar QO frequency (presumed to arise from the electron pocket) and the sign change in the Hall coefficient observed in both families 58 point to a very similar Fermi surface reconstruction. Assuming the presence of additional (though as yet undetected) hole pockets in the reconstructed Fermi surface of Y124 and given that A = 9.5 ± 1.5 nΩ cm K-2, we can estimate using Equation (2) the effective magnitude of the A coefficient associated with an individual -2 hole pocket to be Ah = 49 ± 10 nΩ cm K . Given the strong sensitivity of AKWR to the absolute values of m* and kF, this estimate is considered to be in good agreement with the value of Ah deduced for Y123. This is also in agreement with the two-band description of transport data in Y124 35 and the doping dependence of the Seebeck 20 and Hall 19 coefficients in YBCO. The 2 magnitude of the T term in ρa(T) can thus be considered as yet further evidence that the reconstructed Fermi surface of underdoped Y123 and Y124 contains not only the well- established electron pocket, but also at least one additional hole-like pocket. This is in agreement with the Fermi surface reconstruction scenario proposed within the biaxial CDW model 24,25 , assuming that the initial FS is the pseudogapped FS (e.g. without the states at the anti-node) and not the band structure derived FS. The multi-band scenario has a number of other implications. Firstly, in the Hg1201 family, QOs have been measured at a doping level p = 0.09 with a frequency F = 840 ± 30 T and an effective * -2 mass m = 2.45 ± 0.15 me (ref. 59). Accordingly, Equation (2) gives AKWR = 73 ± 10 nΩ cm K . -2 Above Tc, the measured A coefficient is A ≈ 10 nΩ.cm.K (ref. 34). This large discrepency between the estimated and measured values of A again indicates that the reconstructed Fermi surface of Hg1201 may also contain an additional pocket or pockets that have not yet been observed in QO experiments 60 . Secondly, in Y123, QOs have been observed over a wide range of doping yet F is found to increase only by ~20 % between p = 0.09 and p = 0.152 (ref. 61). These measurements also reveal a strong enhancement of the quasiparticle effective mass as optimal doping is approached and suggest a quantum critical point at a hole doping of pcrit ≈ 0.17. If the electron pocket was indeed the only pocket that persists in the reconstructed phase, this marked enhancement in m* should lead to a corresponding enhancement of the A coefficient on both sides of pcrit , as has been observed, for example, in the transport properties of the isovalently substituted pnictide family BaFe2(As 1-xPx)2 (ref. 38). In cuprates, however, 7 the situation is far from clear. Certainly, there is no sign of a divergence of the A coefficient near p ≈ 0.18 from existing high temperature measurements (ref. 34). Moreover, measurements of the low-temperature in-plane resistivity of several overdoped LSCO samples in high magnetic field have revealed a regime of ‘anomalous’ or ‘extended’ criticality regime around p ≈ 0.19 where the coefficient of the T-linear term is maximal yet there is no sign of divergence or an enhancement in A (from the overdoped side) 5. Within this multi-band scenario, it is possible that the marked enhancement in m* of the electron pocket, and thus in Ae, is offset by changes in Ah (assuming that the hole pockets are always present).

Alternative scenarios Finally, we consider an alternative scenario for the pseudogap in which only Fermi arcs exist at low-field and ask what is the magnitude of the A coefficient expected in such a scenario. This is relatively easy to do, at least approximately. For Y124, p = 0.14. Thus, the full Fermi surface is expected to occupy 57% of the Brillouin zone. If we assume that the quasiparticles on the full Fermi surface have a comparable mass to those found in overdoped cuprates, i.e. m* ~ 5 -2 me, one obtains AKWR = 2.5 nΩ cm K for the full unreconstructed Fermi surface. At low-T, the ‘normal state’ specific heat coefficient γ0 in Y124 is estimated using the entropy conservation construction to be approximately one third of its value at high temperature, i.e. above T* (ref. 52). For a Fermi arc that is one third the length of the full quadrant, AKWR is correspondingly tripled (again assuming an isotropic m*). Thus, the magnitude of A with such a scenario is similar to the coefficient found at low temperatures and from high field studies. It is important to recognize, however, that a Fermi arc has only hole-like curvature, and thus can in no way account for the two-carrier form of the magnetoresistance in Y124, nor for the negative sign of the Hall coefficient at low temperatures and high fields. Until now, all estimates and comparisons have been made under the assumption that the effective mass does not vary around the Fermi surface. In an alternative scenario 62 , the effective mass of the diamond-shaped electron pocket is taken to be anisotropic. The effective mass deduced from quantum oscillations is large because it is dominated by the corner of the pocket which corresponds to the hot spot of the CDW. By contrast, transport is dominated by those regions of the FS with the highest Fermi velocity, i.e. the light quasiparticles in the near-nodal state. This scenario can explain the factor of five discrepancy between the expected value of the KWR ratio and the experimental one (assuming a single electron pocket) but the anisotropy needs to be abnormally strong and would need to increase with doping in order to explain the behavior of the effective mass deduced from quantum oscillations 61 . Similar considerations would also apply if the scattering rate Γ, rather than the effective mass, were strongly anisotropic 63 . The picture of underdoped cuprates now emerging from X-ray spectroscopy, is of a lengthening of the correlation length ξ associated with the charge ordering with increased field strength 15,16 , in agreement with NMR and thermodynamic measurements. At what value of ξ, relative to the mean-free-path of the remnant quasiparticles, does Fermi surface reconstruction appear, or is manifest in the magneto-transport properties, is a crucial open question. At high temperatures, where the mean-free-path is short, the quasiparticle response will be susceptible even to short- range charge-order. At low temperatures, however, the situation is less clear. The findings reported in this manuscript point to a strong influence of the incipient charge order on the transport properties over a wide region of the temperature-doping-magnetic field phase diagram, and calls for a systematic study of the evolution of the low-T in-plane resistivity of underdoped cuprates inside the pseudogap regime.

Acknowledgments We thank K. Behnia, M. Ferrero, A. Georges, and L. Taillefer for useful discussions. Research support was provided by the project SUPERFIELD of the French Agence Nationale de la Recherche, the Laboratory of Excellence “Nano, Extreme Measurements and Theory” in Toulouse, the High Field Magnet Laboratory–Radboud University/Fundamental Research on Matter, and Laboratoire National des Champs Magnétiques Intenses, members of the European Magnetic Field Laboratory.

References

1. Hussey N. E. (2008) Phenomenology of the normal state in-plane transport properties of high-Tc cuprates. J. Phys.: Condens. Matter 20 , 123201. 2. Nakamae S. et al. (2003) Electronic ground state of heavily overdoped nonsuperconducting La 2-xSr xCuO 4. Phys. Rev. B 68 , 100502(R). 3. Mackenzie A. P. et al. (1998) Normal-state magnetotransport in superconducting Tl 2Ba 2CuO 6-δ to millikelvin temperatures. Phys. Rev. B 53 , 5848. 4. Proust. C. et al. (2002) Heat Transport in a Strongly Overdoped Cuprate: Fermi Liquid and a Pure d-Wave BCS Superconductor. Phys. Rev. Lett. 89 , 147003.

5. Cooper R. A. et al. (2009) Anomalous Criticality in the Electrical Resistivity of La 2- xSr xCuO 4. Science 323 , 603-607.

6. Ando, Y. et al ., (2004) Electronic phase diagram of high-Tc cuprate superconductors from a mapping of the in-plane resistivity curvature. Phys. Rev. Lett. 97 , 267001. 7. Ghiringhelli, G. et al. (2012) Long-range incommensurate charge fluctuations in (Y,Nd)Ba 2Cu 3O6+x . Science 337, 821–825. 8. Chang, J. et al. (2012) Direct observation of competition between superconductivity and charge density wave order in YBa 2Cu 3O6.67 . Nature Physics 8, 871–876. 9. Hoffman, J. E. et al . (2002) A four unit cell periodic pattern of quasi-particle states surrounding vortex cores in Bi 2Sr 2CaCu 2O8+ δ. Science 295 , 466–469. 10. Kohsaka, Y. et al . (2007) An intrinsic bond-centered electronic glass with unidirectional domains in underdoped cuprates. Science 315 , 1380–1385. 11. Tabis. W. et al . (2014) Charge order and its connection with Fermi-liquid charge transport in a pristine high-Tc cuprate. Nat. Commun. 5, 5875. 12. Wu, T. et al. (2011) Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa 2Cu 3Oy. Nature 477, 191–194. 13. Wu T. et al . (2014) Emergence of charge order from the vortex state of a high-temperature superconductor. Nat. Commun. 4, 2113. 14. LeBoeuf, D. et al. (2013) Thermodynamic phase diagram of static charge order in underdoped YBa 2Cu 3Oy. Nat. Phys 9, 79–83.

15. Gerber S. et al. (2015) Three-Dimensional Charge Density Wave Order in YBa 2Cu 3O6.67 at High Magnetic Fields. Science 350 , 949-952. 16. Chang J. et al . (2016) Magnetic field controlled charge density wave coupling in underdoped YBa 2Cu 3O6+x . Nature Commun. 7, 11494. 17. Doiron-Leyraud, N. et al. (2007) Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductors superconductor. Nature 447, 565–568. 18. Sebastian S. E. and Proust C. (2015) Quantum Oscillations in Hole-Doped Cuprates. Annu. Rev. Condens. Matter Phys. 6, 411–430. 9

19. LeBoeuf, D. et al. (2011) Lifshitz critical point in the cuprate superconductor YBa 2Cu 3Oy from high-field Hall effect measurements. Phys. Rev. B 83 , 054506. 20. Laliberté, F. et al. (2011) Fermi-surface reconstruction by stripe order in cuprate superconductors. Nature Commun. 2, 432. 21. Forgan E. M. et al. (2015) The microscopic structure of charge density waves in underdoped YBa 2Cu 3O6.54 revealed by X-ray diffraction. Nat. Commun. 6, 10064. 22. Comin R. et al. (2015) Broken translational and rotational symmetry via charge stripe order in underdoped YBa 2Cu 3O6+y . Science 347 , 1335–1339. 23. Sebastian, S. E. et al. (2012) Quantum oscillations from nodal bilayer magnetic breakdown in the underdoped high temperature superconductor YBa 2Cu 3O6+x . Phys. Rev. Lett. 108, 196403. 24. Allais, A., Chowdhury, D. & Sachdev, S. (2014) Connecting high-field quantum oscillations to the pseudogap in the underdoped cuprates. Nat. Commun. 5, 5771. 25. Maharaj, A. V., Hosur, P. & Raghu, S. (2014) Crisscrossed stripe order from interlayer tunneling in hole-doped cuprates. Phys. Rev. B 90, 125108. 26. Doiron-Leyraud N. et al. (2015) Evidence for a small hole pocket in the Fermi surface of underdoped YBa 2Cu 3Oy. Nat. Commun. 6 6034. 27. Maharaj, A. V et al. (2016) Quantum oscillations in a bilayer with broken mirror symmetry: A minimal model for YBa 2Cu 3O6+ δ. Phys. Rev. B 93, 094503. 28. Briffa A. K. R. et al. (2016) Fermi surface reconstruction and quantum oscillations in underdoped YBa 2Cu 3O7-x modeled in a single bilayer with mirror symmetry broken by charge density waves. Phys. Rev. B 93, 094502. 29. Stark, R. W. & Reifenberger, R. (1977) Quantitative theory for the quantum interference effect in the transverse magnetoresistance of pure magnesium. J. Low Temp. Phys. 26 , 763–817. 30. Sebastian S. E. et al. (2010) Fermi-liquid behavior in an underdoped high-Tc superconductor. Phys. Rev. B 81, 140505(R). 31. Grissonnache G. et al. (2016) Wiedemann-Franz law in the underdoped cuprate superconductor YBa 2Cu 3Oy. Phys. Rev. B 93, 064513. 32. Mirzaei S. I. et al. (2013) Spectroscopic evidence for Fermi liquid-like energy and temperature dependence of the relaxation rate in the pseudogap phase of the cuprates. Proc. Natl Acad. Sci. USA 110 , 5774–5778. 33. Rullier-Albenque F. et al. (2007) Total suppression of superconductivity by high magnetic fields in YBa 2Cu 3O6.6 . Phys. Rev. Lett. 99 , 027003. 34. Barisic N. et al. (2013) Universal sheet resistance and revised phase diagram of the cuprate high-temperature superconductors. Proc. Natl Acad. Sci. USA 110 , 12235-12240. 35. Rourke P. M. C. et al. (2010) Fermi-surface reconstruction and two-carrier model for the Hall effect in YBa 2Cu 4O8. Phys. Rev. B 82 , 020514(R). 36. Ando, Y. et al ., (2004) Evolution of the Hall coefficient and the peculiar electronic structure of the cuprate superconductors. Phys. Rev. Lett. 92, 197001. 37. Grissonnanche G. et al. (2014) Direct measurement of the upper critical field in cuprate superconductors. Nat. Commun. 5, 3280.

38. Analytis J. G. et al. (2014) Transport near a quantum critical point in BaFe 2(As 1-xPx)2. Nat. Physics 10 , 194-197. 39. Blanco-Canosa, S. et al. (2014) Resonant x-ray scattering study of charge-density wave correlations in YBa 2Cu 3O6+x . Phys. Rev. B 90, 054513. 10

40. Hücker, M. et al. (2014) Competing charge, spin, and superconducting orders in underdoped YBa 2Cu 3Oy. Phys. Rev. B 90, 054514. 41. Corcoran R. et al. (1994) Quantum oscillations in the mixed state of the type II superconductor 2H-NbSe2. J. Phys.: Condens. Matter 6, 4479. 42. Bel R. & Behnia K. (2003) Ambipolar Nernst Effect in NbSe2. Phys. Rev. Lett. 91 , 066602. 43. Ong N. P. and Monceau P. (1977) Anomalous transport properties of a linear-chain metal: NbSe 3. Phys. Rev. B 16, 3443. 44. Brossard L. et al. (1990) Simultaneous competition and coexistence between charge- density waves and reentrant superconductivity in the pressure-temperature phase diagram of the molecular conductor TTF [Ni(dmit) 2]2. Phys. Rev. B 42, 3935. 45. Abdel—Jawad M. et al. (2006) Anisotropic scattering and anomalous normal-state transport in a high-temperature superconductor. Nat. Phys 2, 821–825. 46. Kadowaki K. and Woods S. B. (1986) Solid State Commun. 58 507. 47. Hussey N. E. (2005) Non-generality of the Kadowaki–Woods Ratio in Correlated Oxides. J. Phys. Soc. Japan 74 , 1107. 48. Cavanagh, D. C. et al ., (2015) Breakdown of the universality of the Kadowaki-Woods Ratio in multi-band metals. Phys. Rev. B 92, 195138. 49. Baber W. G. (1937) The Contribution of the Electrical Resistance of Metals from Collisions between Electrons. Proc. R. Soc. London Ser. A 158 , 383–396. 50. Lin X, Fauqué B., Behnia K. (2015) Scalable T2 resistivity in a small single-component Fermi surface. Science 349 945-948. 51. H. K. Pal, V. I. Yudson, D. L. Maslov. (2012) Lith. J. Phys. 52 , 142–164.

52. Yelland, E. A. et al. Quantum oscillations in the underdoped cuprate YBa2Cu 4O8. (2008) Phys. Rev. Lett. 100 , 047003. 53. Bangura A. F. et al. (2008) Small Fermi surface pockets in underdoped high temperature superconductors: Observation of Shubnikov-de Haas oscillations in YBa 2Cu 4O8. Phys. Rev. Lett. 100 047004.

54. Putzke C. et al. (2016) Inverse correlation between quasiparticle mass and Tc in a cuprate high-Tc superconductor. Science Advances 2 e1501657. 55. Tan B. S. et al. (2015) Fragile charge order in the nonsuperconducting ground state of the underdoped hightemperature superconductors. Proc. Natl Acad. Sci. USA 112, 9568-9572. 56. Riggs, S. C. et al . (2011) Heat capacity through the magnetic-field-induced resistive transition in an underdoped high-temperature superconductor. Nature Phys. 7, 332–335. 57. Marcenat C. et al . (2015) Calorimetric determination of the magnetic phase diagram of underdoped ortho II YBa 2Cu 3O6.54 single crystals. Nature Commun. 6 7927

58. LeBoeuf, D. et al. (2007) Electron pockets in the Fermi surface of hole-doped high-Tc superconductors. Nature 450 , 533–536. 59. Barisic N. et al. (2013) Universal quantum oscillations in the underdoped cuprate superconductors. Nat. Physics 9, 761. 60. Chan M. K. et al. (2016) Single reconstructed Fermi surface pocket in an underdoped single layer cuprate superconductor. ArXiv: 1606.02772 61. Ramshaw B. J. et al. (2015) Quasiparticle mass enhancement approaching optimal doping in a high-Tc superconductor. Science 348 , 317–320. 62. Senthil, T. (2014) On the mass enhancement near optimal doping in high magnetic fields in the cuprates. ArXiv: 1410.2096. 11

63. Robinson P., Hussey N. E. (2015) Ong construction for the reconstructed Fermi surface of underdoped cuprates. Phys. Rev. B 92 , 220501(R).

64. S. Adachi et al . (1998) Preparation of YBa 2Cu 4O8 single crystals in Y2O3 crucible using O2-HIP apparatus. Physica (Amsterdam) 301C , 123-128. 65. Mackenzie, A. P. et al . (1996) Calculation of thermodynamic and transport properties of Sr 2RuO 4 at low temperatures using known fermi surface parameters. Physica 263C, 510- 515. 66. Yoshida T. et al . (2007) Low-energy electronic structure of the high-Tc cuprates La 2−x Sr xCuO 4 studied by angle-resolved photoemission spectroscopy J. Phys. Cond. Matt. 19 , 125209. 67. Hussey N. E. et al. (2013) Generic strange-metal behaviour of overdoped cuprates. J. Phys.: Conf. Ser . 449 , 012004. 68. Rourke P. M. C. et al . (2010) A detailed de Haas–van Alphen effect study of the overdoped cuprate Tl 2Ba 2CuO 6+ δ. New J. Phys. 12 , 105009.

69. Hussey N. E. et al. (1998) Normal-state magnetoresistance of Sr 2RuO 4. Phys. Rev. B 57, 5505-5511. 70. Bergemann C. et al. (2003) Quasi-two-dimensional Fermi liquid properties of the unconventional superconductor Sr 2RuO 4. Adv. Phys. 52 639-725. 71. Nakatsuji S. et al. (2003) Heavy-Mass Fermi Liquid near a Ferromagnetic Instability in Layered Ruthenates . Phys. Rev. Lett. 90 , 137202.

Figure 1. Generic phase diagram of hole doped cuprates. Generic phase diagram of hole-doped cuprates mapped out in terms of the temperature and doping evolution of the in-plane resistivity ρab (T). The solid lines are the phase boundaries between the normal state and the superconducting (SC) or antiferromagnetic (AFM) ground state. The dashed lines indicate crossovers in ρab(T) behaviour. The light blue area corresponds to where incipient charge order (I-CDW) has been detected by X-ray 39,40 measurements in Y123 . The CDW becomes long range order below Tc when a large magnetic field is applied 12-14 .

Figure 2. Field dependence of the in-plane resistivity in Y124. a,b) Electrical resistivity ρa of two samples of YBa 2Cu 4O8 (Tc = 80 K) for current I || a and magnetic field H applied along the c-axis at different temperatures (solid lines). With lowering the temperature, a strong magnetoresistance develops, which can be accounted for by a two-band model 35 . Dashed lines correspond to fits using this model (Equation (1) in the text) to extrapolate the normal-state data to H = 0. 14

Figure 3. Temperature dependence of the in-plane resistivity in Y124. Temperature dependence of the a-axis resistivity of YBa 2Cu 4O8 from which the magnetoresistance has been subtracted using a two-band model to extrapolate the normal-state data to H = 0 for the two samples shown in Fig. 2. Solid lines show the resistivity measured in zero magnetic field. Dashed lines are fits to the Landau-Fermi liquid expectation for the temperature dependence of the resistivity at low temperature, 2 ρa(T) = ρ0 + AT (see text). To estimate error bars we fitted each field sweep data set to Equation (1) between a lower bound Hcutoff and the maximum field strength and monitored the value of ρ(0) as a function of Hcutoff (see Supplementary for more details). 2 The insets are corresponding plot of ρa(T) versus T to highlight the quadratic form of ρa(T) . 15

Figure 4. Fermi surface reconstruction by biaxial charge order. Sketch of the reconstructed Fermi surface adapted from ref. 24 using the CDW wavevectors measured in Y123, showing a diamond-shaped nodal electron pocket (red) and two hole-like ellipses (blue and green). Ae and Ah are the A coefficients estimated from Equation (2) for the electron and hole pockets, respectively. Atot is the expected A coefficient for a parallel resistor model taking into account one electron and two hole pockets and the two CuO 2 planes.

SUPPORTING INFORMATION

1. SAMPLES

3 Single crystalline samples (typical dimensions 400 x 80 x 30 µm ) were flux grown in Y 2O3 crucibles in a partial oxygen pressure of 400 bar [64]. In contrast to the Y123 family, which has a single CuO chain with variable oxygen content, Y124 contains alternating stacks of CuO 2 bilayers and double CuO chains (along b-axis) that are stable and fully loaded.

2. MEASUREMENT OF THE LONGITUDINAL AND TRANSVERSE RESISTANCES

The pulsed-field measurements were performed using a conventional 6-point configuration with a current excitation between 5 mA and 10 mA at a frequency of ~ 50 kHz. A high-speed acquisition system was used to digitize the reference signal (current) and the voltage drop across the sample at a frequency of 500 kHz. The data were post-analyzed using software to perform the phase comparison. Data for the rise and fall of the field pulse were in good agreement, thus excluding any heating due to eddy currents.

3. TWO-BAND MODEL

Assuming that the Fermi surface of underdoped YBa 2Cu 3Oy contains both electron and hole pockets, the transverse magnetoresistance can be fitted with a two-band model:

(σ + σ )+ σ σ (σ R2 + σ R2 )H 2 α H 2 ρ(H ) = h e eh hh ee = ρ + ()()σ + σ 2 + σ σ 22 + 2 2 0 + β 2 h e eh Rh Re H 1 H (S1) 1 Where ρ = (S2) 0 σ +σ h e (σ +σ )σ σ (σ R2 +σ R2 )−σ 2σ 2 (R + R )2 α = h e eh hh ee h e h e (S3) ()σ +σ 3 h e σ 2σ 2 (R + R )2 β = eh h e (S4) ()σ + σ 2 h e

σh (σe) is the conductivity of holes (electrons) and Rh (Re) is the Hall coefficient for hole (electron) carriers. Using the three free fitting parameters ρ0, α and β, we were able to subtract the orbital magnetoresistance from the field sweeps and get the temperature dependence of the extrapolated zero-field resistivity ρ(0 ), which is the main focus of the present study. A detailed description and discussion of α and β, meanwhile, is considered beyond the scope of the present paper. 17

The choice of this model is justified by examining closely the form of the magnetoresistance at high temperatures above Tc. In Fig. S1, we compare the fit using the two-band model with a simple H2 magnetoresistance. At T = 100 K where there is no trace of superconductivity, it is evident that a simple H2 magnetoresistance cannot fit the data whereas the two-band model provides a reasonable fit to the data over the entire field range.

4. FIT OF THE MAGNETORESISTANCE To track the temperature dependence of the zero-field resistivity ρ(0), we fitted each field sweep data set between some variable lower bound Hcut-off and Hmax , the peak field of each pulse, to the two-band form and monitored the value of ρ(0) as a () = (0) + function of Hcut-off . Example plots of ρ(0) versus each given Hcut-off are shown in Figure S2 (open symbols). For Hcut-off less than some critical field, defined as H*, the extracted ρ(0) values rise monotonically with increasing cut-off field as one enters the resistive transition. For Hcut-off > H*, ρ(0) reaches a plateau value, indicating that the magnetoresistance now follows the two- band form shown above. As the field range for fitting becomes too narrow, the extrapolated value of ρ(0) begins to oscillate wildly (not shown). Having determined H*, we then fitted each field sweep data set between H* (now fixed) and some variable upper bound Hupper (full symbols in Fig. S2 and S3).

At all temperatures, the field range to obtain reliable values of ρ(0) is of the order of 10 – 15 * T or more concretely, between 1.3 H < H < Hmax . Indeed, if we look to the curve at T = 60 K (Hmax = 55 T), for example, the value of ρ(0) changes only by a few percent when changing the upper cut-off from 55 T down to 40 T. In another words, fitting the data between H* = 30 T and Hupper = 40 T gives the same value of ρ(0) to within ± 5 % as fitting between 30 T and 55 T. We used the scatter in the value of ρ(0) when fitting the data with a upper cut-off to define the error bar (standard deviation).

At temperatures lower than T = 20 K, we noticed that all curves merge on top on the others for sample #1 (see Fig. S4) showing that the magnetoresistance and the extrapolated resistivity at H = 0 cannot be differentiated within our experimental uncertainty. For sample #2, due to a smaller maximum field at T = 20 K and to a slight temperature dependence of the magnetoresistance, the error bars are higher below T = 20 K.

5. EFFECTIVE MASS AND SPECIFIC HEAT From the measured effective mass m*, the residual linear term γ in the electronic specific heat Ce(T) at T → 0 can be estimated through the relation [65]

2 * γ = (1.46 mJ / K mol) Σi (ni mi / m0)

th where ni is the multiplicity of the i type of pocket in the first Brillouin zone (This expression assumes an isotropic Fermi liquid in two dimensions with a parabolic dispersion). For a Fermi surface containing one electron pocket and two hole pockets per CuO 2 plane, we obtain a total 2 mass of (1.4± 0.1) + 2 (0.45 ± 0.1) = 2.3 ± 0.3 m0, giving γ = 6.7 ± 0.9 mJ / K mol (for two CuO 2 planes per unit cell). High-field measurements of Ce at T → 0 in YBCO at p ~ 0.11 yield γ = 6.5 18

2 ± 1.5 mJ / K mol [57] at H > Hc2 = 30 T. Note that this measured value includes a residual electronic specific heat contribution that is present even in zero-field. We therefore find that the Fermi surface of YBCO can contain at most two small hole pockets in addition to only one electron pocket per CuO 2 plane. No further sheet can realistically be present in the Fermi surface.

6. KADOWAKI-WOODS ESTIMATION IN CORRELATED SYSTEMS

Comparison of AKWR with experimentally determined values shows good agreement for both single- and multi-band correlated systems. In the single-band, highly overdoped LSCO ( p = -2 -2 0.33) for example, A = 2.5 ± 0.5 n Ω cm.K [2]. This compares with AKWR = 4 + 1 nΩ cm.K (using * -1 m = 4.7 + 1.0 me and kF = 5.5 + 0.2 nm from specific heat [2] and ARPES measurements [66]). Similarly, in highly overdoped Tl2201 ( p = 0.3), A = 5.4 ± 0.5 n Ω cm.K-2 [67], while from detailed * -1 quantum oscillation measurements (m = 5.2 + 0.5 me and kF = 7.4 + 0.1 nm [68]), one obtains -2 AKWR = 3.9 + 0.4 nΩcm.K .

In multi-band systems, one may assume that bands contribute in parallel, e.g. 1/ A = ∑i 1/ Ai. In -2 Sr 2RuO 4, for example, A = 5 + 1.5 n Ω cm.K [69]. Taking the values for m* and kF for the three -2 Fermi cylinders deduced from QO measurements [70], one obtains AKWR = 3.6 + 0.5 nΩ cm.K , again in good agreement with the measured value. In isovalently substituted Ca 2-xSr xRuO 4, the system goes from a three-band metal in the Ca-free case (at x = 2) to a single-band metal for x < 0.5 as the so-called α and β bands become localized [71]. This crossover from a single- to a multi-band system leads to an increase in the KWR of three orders of magnitude [71] that can be explained quantitatively using Equation (2) of the main text. With regards to the hole-doped cuprates, it is questionable whether the KWR should hold at all within the pseudogap phase. However, given the increasing amount of data pointing to a rather conventional state at sufficiently low temperature, we believe that such analysis and comparison is appropriate. Thus, we are confident that the approach described here is also applicable to the reconstructed phase of underdoped cuprates.

Table S1: List of previous reports of T 2 resistivity in underdoped cuprates

Compound Doping level T-range of T 2 resistivity Reference

YBCO 0.03 140 K to 300 K [36] YBCO 0.09 60 K to 150 K [33] Hg1201 0.055 100 K to 190 K [34] Hg1201 0.075 85 K to 220K [34] Hg1201 0.1 90 K to 170K [34] LSCO 0.01 180 K to 300 K [34] LSCO 0.02 140 K to 250 K [36] LSCO 0.08 60 K to 160 K [36]

Table S2: Values and error bars of the parameters of the two-band model for sample #1:

Temperature[K] ρ(0) [µ Ω cm] α [10−3 µΩ cm T -2] β [10−3 T−2]

80 60.6 ± 0.8 23.8 ± 7 1.1 ± 0.5

70 50.9 ± 0.8 20.75 ± 3 0.64 ± 0.12

60 42.7 ± 1.0 14.4 ± 2.3 0.3 ± 0.08

50 34.7 ± 1.4 14.3 ± 2.3 0.2 ± 0.07

40 26.3 ± 2.0 16.7 ± 3 0.14 ± 0.06

34 18.7 ± 1.0 22.7 ± 1.3 0.17 ± 0.02

27 12.5 ± 1.0 27.4 ± 1.1 0.17 ± 0.01

20 8.7 ± 3.0 33 ± 3 0.22 ± 0.03

12 8.7 ± 3.0 33 ± 3 0.22 ± 0.03

4.2 8.7 ± 3.0 33 ± 3 0.22 ± 0.03

1.5 8.7 ± 3.0 33 ± 3 0.22 ± 0.03

Table S3: Values and error bars of the parameters of the two-band model for sample #2:

Temperature[K] ρ(0) [µ Ω cm] α [10−3 µΩ cm T -2] β [10−3 T−2]

64.1 ± 1.0 12.8 ± 1.4 0.36 ± 0.07

70 52.5 ± 0.6 13.3 ± 0.7 0.28 ± 0.03

60 41.0 ± 0.6 13.7 ± 0.6 0.22 ± 0.02

50 31.0 ± 0.8 14.8 ± 0.7 0.19 ± 0.02

40 22.3 ± 1.4 19.7 ± 1.1 0.22 ± 0.02

30 14.9 ± 1.4 26.3 ± 1.1 0.26 ± 0.01

20 12 ± 2 30 ± 5 0.29 ± 0.05

10 12 ± 5 30 ± 10 0.29 ± 0.1

4.2 12 ± 5 30 ± 10 0.29 ± 0.1

1.5 12 ± 5 30 ± 10 0.29 ± 0.1 20

Figure S1: Comparison of the fit using a two-band model and a H2 magnetoresistance at T = 100 K (sample #1).

Electrical resistivity ρa of YBa 2Cu 4O8 (sample #1) at T = 100 K (black line). The red curve corresponds to a fit to the data using the two-band expression () = (0) + . For the blue curve, the low-field data were fitted to the single-band form () = (0)+.

Figure S2: Mode of extraction of the zero field resistivity ρ(H = 0) for Y124 (sample #1) at different temperatures.

The data were first fitted to the form between some lower () = (0) + bound Hcut-off and the peak field of each pulse and a value for ρ(0) extracted (open symbols). The arrows indicate the value of the field strength H* below which the extrapolated ρ(0) value decreases rapidly as one enters the resistive transition. The data were then fitted to the same expression between H* and some upper bound Hupper which is itself then varied. The values of ρ(0) deduced from this procedure (full symbols) are shown plotted as a function of Hupper , the sample held at the various temperatures indicated. These values indicated by the solid symbols are then averaged to determine ρ(0) for each pulse.

Figure S3: Estimation of the error bars of ρ(0) for sample #1

Same as Fig. S2 with only the extrapolated ρ(0) values obtained from the fit between H* and Hupper shown for clarity. The solid horizontal lines mark the mean value for ρ(0) at each temperature while the dashed lines indicate the corresponding error bars.

Figure S4: Field dependence of the in-plane resistivity in Y124 (sample #1) at low temperatures.

Electrical resistivity ρa of YBa 2Cu 4O8 (sample #1) between T = 1.5 K and T = 20 K, showing that the magnetoresistance and the extrapolated resistivity at H = 0 cannot be differentiated below 20 K within our experimental uncertainty. The dashed line corresponds to the two-band fit at T = 20 K used to extrapolate the normal-state data to H = 0.

Figure S5: Mode of extraction of the zero field resistivity ρ(0) and error bars for sample #2 at the different temperatures indicated.

n Figure S6: Fits of the resistivity to a power law ρa(T) = ρ0 + AT .

Temperature dependence of the a-axis resistivity of YBa 2Cu 4O8 from which the magnetoresistance has been subtracted using a two-band model to extrapolate the normal-state data to H = 0 for the two samples shown in Fig. 2. Solid lines show the resistivity measured in zero magnetic field. Dashed lines are fits to a power law ρa(T) = ρ0 + AT n where the exponent n is indicated in the figure for each sample. Note that the points above 60 K have been omitted for the fit of sample #1.